Friday, December 20, 2013

Solving Hilbert’s sixth problem (part one of many)

Outside in or inside out?

In 1900 David Hilbert proposed a set of problems to guide mathematics in the 20th century. Among them problem six asks for the axiomatization of physics.

Solving problem six is a huge task and the current consensus
is that it is a pseudo-problem but I will attempt to prove otherwise in this
and subsequent posts. I will also start formulating the beginning of the
answer.

Let’s first try to get a feel for the magnitude of the
problem. What does axiomatizing physics mean? Suppose the problem is solved and
we have the solution on a piece of paper in front of us. Should we be able to
answer any physics question without using experiments? Is the answer
supposed to be a Theory of Everything? Let’s pause for a second and reflect on
what we just stated: eliminate the need for experiments in physics!!! This is
huge.

But what about Gödel Incompleteness Theorem?
Because of it mathematics is not axiomatizable and has an infinite landscape.
Do the laws of physics have an infinite landscape too?

The biggest roadblock for solving Hilbert’s sixth problem
turns out to be Gödel Incompleteness Theorem. Let’s get the gist of it. Start
with an antinomy (any antinomy will do): this
statement is false. If the statement is true, its content is accurate but
its content says that the statement is false. Contradiction. Likewise, if the
statement is false, its negation is true, but the negation states that the statement
is true. Again we have a contradiction. This was well known a long time before
Gödel as the liars’ paradox. But now let’s follow Gödel and replace true and
false with provable and unprovable. We get: this
statement is unprovable. Suppose the statement is false. Then the statement
is provable. Then there exists a proof to a false statement. Therefore the
reasoning system is inconsistent. The only way to restore consistency is to
have that the statement is true. Hence we just constructed a true but unprovable
statement!

Now in a sufficiently powerful axiomatic system Sn suppose we
start with axioms: a1, a2, …, a_n (at the minimum Sn must include the natural number
arithmetic). Construct a statement P not provable in the axiomatic system
(Gödel does this using the diagonal argument).
Then we can add P to a1,…,a_n and construct the axiomatic system S_n+1 = a1,
a2, …, a_n, P. We can also construct another axiomatic system S’_n+1 = a1, a2,
…, a_n, not P. Both S_n+1 and S’_n+1 are consistent systems, but together are
incompatible (because P and not P cannot be both true at the same time). The
process can be repeated forever, and hence in mathematics there is no “Theory
of Everything Mathematical”, no unified axiomatic system, and mathematics has
an infinite domain.

So it looks like the goal of axiomatizing physics is
hopeless. Mathematics is infinite, and mathematicians seem to be able to keep
exploring the mathematical landscape. Since mathematicians are part of nature
too, axiomatizing physics seem to demand math axiomatization as well. Case
closed, Hilbert sixth problem must be a pseudo-problem, right?

However, it turns out there is another way to do
axiomatization. Let’s start by looking at nature. We see that space-time is
four dimensional, we see that nature is quantum at core, we see that the
Standard Model has definite gauge symmetries. Nature is written in the language
of mathematics. But WHY some
mathematical structures are preferred by
Nature over others? We cannot say that those mathematical structures
are unique, all mathematical structures are unique! We can say that some mathematical structures are
distinguished.

Solving Hilbert sixth
problem demands as a prerequisite finding a mechanism to distinguish a handful
of mathematical structures from the infinite world of mathematics.

And in a well known case we know the answer. Consider the special
theory of relativity: this is a theory based on a physical principle. Finding essential physical principles is what needs
to be done first. Suppose we now have all nature’s physical principles
written in front of us. What is the next step? The next step is to use them as filters to select distinguished
mathematical structures. If we pick the principles correctly, the accepted mathematical
structures will be those and only those which are distinguished by nature as
well.

So instead of doing an axiomatization in the traditional way
outside in: from axioms
derive statements, we use it inside
out: from physical principles (axioms) we reject all mathematical
structures but a distinguished few. The axioms are like the fence of a
domain establishing its boundary. The orientation of the boundary matters. In
everyday life, or in engineering this way of using the axioms is well known:
they are called requirements. When we buy a car we do not start with the
Standard Model Lagrangean to arrive at the make and model we will buy, but we
start with requirements: the price range, the acceptable colors, etc. In other
words we start with the acceptable features. In special theory of relativity,
the relativity principle rejects all Lie groups except the Lorenz and the Galilean
group. An additional constant of the speed of light postulate picks the final
answer.

Whatever gets selected does not need to be a closed form
theory of everything and we bypass the limitation from Gödel incompleteness
theorem. Now this program is actually very feasible. Next time I will show how
to pick the physical principles, we’ll pick two principles and in subsequent
posts I’ll use those principles to derive quantum and classical mechanics step
by step in a very rigorous mathematical way (it is rather lengthy to derive
quantum mechanics and I don’t know how many posts I’ll need for it). It turns
out that quantum and classical mechanics are also theories of nature based on
physical principles just like theory of relativity is. The role of the constant
of the speed of light postulate will be played in the new case by Bell’s
theorem. In the process of deriving quantum mechanics we’ll make great progress
towards solving Hilbert’s sixth problem, but we’ll still be far short from a
full “theory of everything”.

2 comments:

"Solving Hilbert sixth problem demands as a prerequisite finding a mechanism to distinguish a handful of mathematical structures from the infinite world of mathematics." The mechanism might be the physical interpretation of string theory. I believe I have found the interpretation.

David, In one of your papers you state: "G. ‘t Hooft, Han Geurdes, J. Christian, and many other theorists have worked on refutations of Bell’s theorem and/or the replacement of quantum theory by some deterministic theory"

I devoted an extensive amount of time to understand what J. Christian claims, and I can very confidently state that all those claims are incorrect. See http://arxiv.org/abs/1109.0535 J Christian makes a simple mathematical error of +1= -1. The mistake was discovered independently by 4 people, 3 wrote papers about it, one (who was first to notice it) did not publish, but let J. Christian know about it privately, and I was first to publish it. A panel of experts by FQXi later on confirmed the mistake.